How to Assemble Tetrahedrons from Building Blocks:

when the building blocks are made of closest packed spheres

It is simple to assemble tetrahedrons of different sizes when the building block is one sphere, but it is quite different if we try to assemble with bigger building blocks. The pictures below illustrate the building block composition of the tetrahedrons as they increase in size.

(click on pictures to enlarge)

A        B           C              D                   A1                       B1...

Fig. 1  Tetrahedral growth

Fig. 2   Exposed centers

Fig. 3   Octahedral growth

Fig. 4   One-sphere growth
(conventionally known as cuboctahedral)

Note: In order to make it easier to understand the patterns, I decided to use the polyhedral nomenclature to describe the building blocks - such as the tetrahedron and octahedron, even though building blocks do not behave as polyhedrons. 

Shown in Fig.1 is a series of tetrahedrons that are growing in size. The edges of the tetrahedrons are increasing by one sphere at a time. 

In Fig. 2 the top and front tetrahedrons have been removed so the centers can be seen.


As you can see in Fig.1, the composition of the tetrahedrons are size dependent. The building blocks differ as the size of the tetrahedrons increase. There are several recursive patterns.

What can we tell about these series of structures?

1. The overall repeating pattern in (Fig. 1) is that of the first four tetrahedrons labeled A, B, C, and D, which repeats  as it grows bigger - A1, B1, C1, D1,  A2,  B2, ... . The centers of each of the four tetrahedrons (A, B, C, D) are different building blocks, and they are color coded white, black, blue and red, as shown in (Fig. 1, 2, 3)
2. Embedded in the tetrahedral growth series are also the growth of the octahedral and the one-sphere (cuboctahedral) growth series (Fig. 3, 4). 
3. The relationship between the 4 center building blocks is that of 2 pairs of duals that are evolving (double duals): A-C and B-D. White A (up-tetrahedron) combines with blue C (down-tetrahedron) and black B (octahedron) combines with red (one-sphere, cuboctahedron). As shown in (Fig. 1) the 2 pairs of duals alternate in the progression and the duals themselves also alternate. Duals A-C are even number edge-spheres tetrahedrons and duals B-D are odd numbers edge-spheres tetrahedrons.
4. In Fig. 1, every other tetrahedron (B, D, B1, D1...) have the familiar pattern of the octet (octahedron-tetrahedron) combination. Note that the octahedrons are always one-edge sphere bigger than the complementary tetrahedron. It is not equal in edge length as in the octet truss.
5. The other half of the series is composed of 'up'-tetrahedron and its complementary 'down'-tetrahedron. As fig. 2 shows - the first white tetrahedron (A) is pointing up, and in the third (C) the blue tetrahedron in the center is pointing down, next in (A1) the tetrahedron in the center is white (up) again, and it keeps alternating in this manner.
6. These 4 building blocks re-occur in the 4 isotropic vector matrix (4 fcc lattices) in "Mapping the Hidden Patterns in Sphere Packing" R Chu 2003 Chart 1 .
7. These patterns are like 3dimensional fractals. We can substitute the octahedron in D (Fig. 1) with smaller building blocks or substitute the tetrahedrons in A1 (in Fig. 1) in the manner shown in (Fig. 5) below. As the tetrahedrons get bigger we can progressively substitute with smaller equivalent building blocks. This process of substituting with smaller building blocks could continue until it is composed of only single sphere building blocks.
8. Fig. 4 shows the one-sphere growth surrounded by 12 spheres in the second layer (cuboctahedron). It is Fuller's vector equilibrium (VE). 
9. All the possible building block combinations of tetrahedrons are the  symmetries within the tetrahedron. 

Fig. 5 

In Fig. 5, the 5sphere-edge tetrahedron and the 6sphere-edge tetrahedron show an increased level of complexity compared to D and E1 in (Fig. 1). As the tetrahedrons increase in size the levels of complexity increases.  So far we have only been looking at closest packed building blocks. A great variety of lesser density packing can be considered.

I do not have the skills to write these patterns as fractals or describe them mathematically. If any one is interested I would be happy to collaborate. Comments or questions are welcomed.
Russell Chu, 206-313-5708 (verbchu@gmail.com)